A Pollard Type Result for Restricted Sums

Abstract

Let be an arbitrary field. Letpbe the characteristic of in case of finite characteristic and [infinity] if has characteristic 0. LetAbe a finite subset of . By [logical and]2 Awe denote the set {a+b a, b[set membership, variant]Aanda[not equal to]b}. Forc[set membership, variant][logical and]2 A, let[nu](R)cbe one-half of the cardinality of the set of pairs (a, b) satisfyinga[not equal to]banda+b=c. Denote by[mu](R)ithe cardinality of the set {c[set membership, variant][logical and]2 A [nu](R)c[greater-or-equal, slanted]i}. We prove that, fort=1, ..., [left floor]A/2[right floor], [summation operator]ti=1 [mu](R)i[greater-or-equal, slanted]t min{p, 2(A-t)-1}. For =0pandt=1 we get the Erdos-Heilbronn conjecture, first proved by J. A. Dias da Silva and Y. O. Hamidoune (Bull. London Math. Soc.26, 1994, 140-146).